Higher differentiability results for solutions to a class of non-homogeneouns elliptic problems under sub-quadratic growth conditions

Abstract

We prove a sharp higher differentiability result for local minimizers of functionals of the form $$\mathcal{F}\left(w,\Omega \right)={\int }_{\Omega }\left[F\left(x,Dw(x)\right)-f(x)\cdot w(x)\right]dx$$ with non-autonomous integrand $F(x,\xi )$ which is convex with respect to the gradient variable, under $p$-growth conditions, with $1<p<2$. The main novelty here is that the results are obtained assuming that the partial map $x\mapsto D_{\xi }F(x,\xi )$ has weak derivatives in some Lebesgue space $L^q$ and the datum $f$ is assumed to belong to a suitable Lebesgue space $L^r$. We also prove that it is possible to weaken the assumption on the datum $f$ and on the map $x\mapsto D_{\xi }F(x,\xi )$, if the minimizers are assumed to be a priori bounded.